Last modified date: <%+ tp.file.last_modified_date() %>
- Tags: - #todo/untagged - Refs: - UROP overview: https://math.mit.edu/research/undergraduate/urop-plus/documents/2020/Benson.pdf#page=1 #resources/notes #resources/summaries - Links: - compactly generated - SHC
tensor triangulated category
A tensor-triangulated category is a triple \((\mathcal{K}, \otimes, 1)\) consisting of a triangulated category \(\mathcal{K}\), a symmetric monoidal product \(\otimes: \mathcal{K} \times \mathcal{K} \rightarrow \mathcal{K}\) which is exact in each variable.
Thickness
Given a \(\mathsf{T}\in {\mathsf{triang}}\mathsf{Cat}\), a thick subcategory \(S\) is a full subcategory of \(T\) which is closed under finite direct sums and summands.
Spectrum
- A subcategory \(I\leq \mathsf{T}\) is an ideal iff whenever \(i\in I\) and \(k\in \mathsf{T}\) is compact, \(k\otimes i\in I\).
- A proper thick ideal \(X\leq \mathsf{T}\) is prime iff \(x\otimes y\in I \implies x\in I\) or \(y\in I\).
- The spectrum of a thick ideal is \(\operatorname{Spc}(I) = \left\{{I\leq \mathsf{T} {~\mathrel{\Big\vert}~}I \text{ is prime}}\right\}\).
Localization
- A subcategory \(S \subset T\) is localizing if it is thick and closed under small coproducts.
- A subcategory colocalizing if it is thick and closed under small products.
SHC
